Note on a classical case of self-organization: The Persian Immortals

In an article in this journal, Kanellopoulos, Razis, and van der Weele provided a key formula that describes the evolution of the cumulative distribution of the Persian Immortals' fighting abilities during a battle where fallen soldiers, who are more likely to be weaker than average, are replaced by reserve soldiers with average strength. They also calculated the limit distribution, which shows the final strength advantage of the fighting army over the reserve unit. However, their arguments to derive the mentioned key formula may be difficult to follow for some readers who are not familiar with how the cumulative distribution function of a set changes when some of its elements are replaced. Here, we present a simpler yet more general method to derive this formula. This makes Kanellopoulos, Razis, and van der Weele's compelling exercise more appropriate for an introductory undergraduate course.